Questions

01

Even and odd parts of a Fourier series

Let $f(x)=\frac{1}{2}{a}_0+{\sum }_{n=1}^{\infty }(a_n\cos (nx)+b_n\sin (nx))$.

Prove that the even part $f^e(x)$ is the cosine series, while the odd part $f^o(x)$ is the sine series.

02

Establishing Fourier orthogonality

Show that ${\int }_{-\pi }^{\pi }\cos (nx)\cos (mx)dx=\{\begin{array}{ll}\pi & n=m\\ 0& n\ne m.\end{array}$
Hint: average the trig identities for $\cos (A+B)$ and $\cos (A-B)$.